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INVENTORY MODEL WITH PLANNED SHORTAGES 421

10.4 Inventory Model with Planned Shortages

A shortage or stock-out is a demand that cannot be supplied. In many situations,
shortages are undesirable and should be avoided if at all possible. However, in other
cases it may be desirable – from an economic point of view – to plan for and allow
shortages. In practice, these types of situations are most commonly found where the
value of the inventory per unit is high and hence the holding cost is high. An example
of this type of situation is a new car dealer’s inventory. Often the specific car that a
customer wants is not in stock. However, if the customer is willing to wait a few weeks,
the dealer is usually able to order the car for delivery at some stage in the future.
The assumptions of the The model developed in this section takes into account a type of shortage known
EOQ model in Table 10.3 as a backorder. In a backorder situation, we assume that when a customer places an
apply to this inventory
model with the exception order and discovers that the supplier is out of stock, the customer waits until the new
that shortages, referred shipment arrives, and then the order is filled. Frequently, the waiting period in back-
to as backorders, are ordering situations is relatively short. So, by promising the customer top priority and
now permitted.
immediate delivery when the goods become available, companies may be able to
convince the customer to wait until the order arrives. In these cases, the backorder
assumption is valid.
The backorder model that we develop is an extension of the EOQ model pre-
sented in Section 10.2. We use the EOQ model in which all goods arrive in inventory
at one time and are subject to a constant demand rate. If we let S indicate the
number of backorders that are accumulated when a new shipment of size Q is
received, then the inventory system for the backorder case has the following char-
acteristics:
l If S backorders exist when a new shipment of size Q arrives, then S backorders
are shipped to the appropriate customers and the remaining Q S units are
placed in inventory. Therefore, Q S is the maximum inventory.
l The inventory cycle of T days is divided into two distinct phases: t 1 days when
inventory is available and orders are filled as they occur, and t 2 days when
stock-outs occur and all new orders are placed on backorder.
The inventory pattern for the inventory model with backorders, where negative
inventory represents the number of backorders, is shown in Figure 10.6.
With the inventory pattern now defined, we can proceed with the basic step
of all inventory models – namely, the development of a total cost model. For

Figure 10.6 Inventory Pattern for an Inventory Model with Backorders

Maximum
Q – S
Inventory
Inventory 0 Time

– S
t 1 t 2
T

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